TODAY, IT IS almost a cliche to describe the rise of 3-D printing
as a groundbreaking development. The notion that the technique
represents a decisive turning point in the history of technology has
gained widespread acceptance, with oh-so-grand pronouncements of its
power coming from the likes of Barack Obama, Elon Musk, even Martha
Stewart. And as the technology has become increasingly accessible and
widely adopted in the intervening years, the vision of a 3-D-printed
world seems less like science fiction than like a rapidly approaching
reality. Indeed, "Mutations- Creations/ Imprimer le monde"
(Mutations-Creations / Print the World), a major exhibition opening at
the Centre Pompidou in Paris this month, offers just such a proposition,
showcasing 3-D- printed products that are purportedly transforming a
wide range of fields. Meanwhile, efforts are under way to adapt the
technology to seemingly endless manufacturing applications, for the
creation of everything from clothing to body parts. At the architectural
scale, house prototypes are already being 3-D-printed from concrete, and
in 2012 Foster + Partners unveiled a design for a moon colony to be
built by space-traveling robots equipped with 3-D-printing arms.

Yet all this emphasis on making, on the physical results of the
3-D-printing process, threatens to obscure a far more fundamental
revolution. Three-dimensional printing is only the tip of the iceberg,
just one of a host of digital techniques for scanning, visualization,
and modeling that profoundly alter how we make things--but also how we
understand and represent the world around us, how we see and what we
experience. Until now, modern culture and technology have been largely
image-based; images, in other words, were the primary means of recording
and transmitting information about the world. But 3-D printing is merely
the most visible symptom of a paradigm shift in global technology and
culture from the visual to the spatial. And to fully grasp the magnitude
of this change, we must look not only forward but back: to the entire
development of technologies of reproduction and representation since the
Renaissance. Here, Artforum invited architectural historian Mario Carpo
to reflect on both the historical roots and the future implications of
this looming technological revolution.

THE PROGRESS of contemporary digital technologies over the course
of the past thirty years--from verbal to visual to spatial
media--curiously reenacts, in a telescoped time line, the entire
development of Western cultural technologies from the beginning of
recorded history. (1) One generation ago, digital tools dealt almost
exclusively with alphanumeric data; then came the rise of digital
images, and today digital tools can easily manipulate all kinds of
volumes in three dimensions. This is because words use less data than
images, and flat, planar images use less data than volumes in space; as
computers became more powerful and cheaper, digital notations could move
from the alphabet to pixels to voxels. (2)

The same sequence already occured over a much longer period:
Throughout classical antiquity and the Middle Ages, the main vehicle for
the recording and transmission of data was verbal, not visual. Words
were recorded and transmitted in space and time using the technology of
the alphabet, but images were not. Classical and medieval authors had
good reasons not to trust images: First, classical antiquity bequeathed,
and likely knew, no geometric rules for making drawings-- rules whereby
every artist looking at the same thing would make the same drawing, and
everyone looking at the same drawing would see the same thing. Second,
no technology existed to make identical copies, so that each iteration
of any given drawing was at the mercy of the will of individual
copyists.

These two conditions changed suddenly and drastically in the
Renaissance, due to the near-simultaneous emergence of perspective and
of xylographie printing. Leon Battista Alberti's perspective (first
described in his treatise Delia pittura |On Painting], 1435)
standardized the way images are captured: Once the geometry of a
snapshot of the world is set by choosing the vantage point and the
direction of the central visual ray, the resulting image is always the
same--never mind if it is made by me, by you, or by a mechanical camera.
This is because that image is a geometric projection, and Alberti's
rules explain how to make that projection, and how to notate it once and
for all. Print, in turn, standardized the way images are reproduced.
Once a drawing is engraved on a mechanical matrix and printed, every
copy of it will look the same. From their capture to their
dissemination, modern images thus acquired a double guarantee of
trustworthiness: true to nature when drawn by the artist; true to the
artist's drawing when reproduced by the printer. These were, at
long last, images that everyone could use and trust, and so everyone
did: After many centuries of undisputed dominion of the word in the
Renaissance, Western culture went visual, (3) and the dominion of the
eye has since left an indelible mark on all aspects of Western
modernity. But the cultural and technical primacy of modern,
perspective, projected images--and, with that, of images in general--is
now drawing to a close. The imminent demise of the image-based visual
culture of modernity is all the more inevitable as it is due not to
ideology, but to sheer technological obsolescence: Just as during the
Renaissance, Western information technologies went from verbal to
visual, today global technology and culture are going from visual to
spatial--from 2-D to 3-D, from perspectival projection to volumetric
point cloud.

RENAISSANCE ARTISTS were keenly aware of the technological novelty
of the images they were using, and of the advantages they offered.
Painting came to be seen as equal to the written word, even competing
with poetry. (4) Painters, who were manual workers and guild members
like all others in the Middle Ages, started to be seen as artists,
dealing as they were with a new class of high-tech artifacts:
perspectival images. But some Renaissance artists were also sculptors,
and the competition between painting and sculpture (the "paragone
delle arti"), or between 2-D and 3-D copies, soon became one of the
hottest topics of art theory. The dispute culminated around the
mid-sixteenth century, when the Florentine humanist and historiographer
Benedetto Varchi (1503-1565) posted a call for papers on the subject and
published the replies he received in a volume, preceded by his own
lengthy essay. (5) Michelangelo, the only contributor Varchi cites on
the title page, unsurprisingly championed sculpture. Always a revolte,
Michelangelo was going against the stream. Almost everyone else in the
Renaissance rooted for painting.

The main arguments for the supremacy of painting over sculpture had
already been staked by Leonardo da Vinci around 1492. (6) Sculpture is a
craft, and sculptors are manual workers, whereas painting is based on
the mathematical laws of perspective, hence the painter is a
mathematician and a scientist. Sculpture may be closer to reality in a
literal sense, but perspectival images do not only represent reality,
they also measure it. By the way they are made, perspectival images
embed the precise proportional measurements of whatever they show,
because all they show has been measured by the painter and can be
measured again in the painting by each viewer. This is because, in
Leonardo's words, "perspective is a very subtle invention and
investigation of mathematical studies," based on "laws and
demonstrations." (7) In today's terms, the geometry of the
perspectival construction is reversible: Alberti's geometric rules
convert every point in space, including infinity, into a point of the
picture plane, and the other way around--or almost, as in all
projections. (8) In short, the main advantage of painting (9) over
sculpture is scientific precision: Perspective is a measuring tool--as
much a tool of representation as a tool of quantification. Yes,
perspectival images also look quite similar to the things we see, but if
realism had been the only criterion, sculpture would easily have won the
day, as sculpture is much more similar to three-dimensional reality than
any planar image can ever hope to be.

Indeed, even Leonardo had been obliged to admit that sculpture can
better represent an object in the round that any single painting can, as
sculptures offer views of an object from all vantage points, and a
picture is limited to one. Leonardo's counterargument was that two
pictures, each composed from a well- chosen vantage point--back and
front, for example--can capture enough data to describe any 3-D object
in full, but he must have known that that is not always the case. (10)
Starting with Lorenzo Lotto, Renaissance painters sometimes provided
full identification of their subjects by combining three, not two, views
in the same painting. Lotto appears to have rotated his subject by
approximately 120 degrees at a time, thus offering a partial view from
the back. When Van Dyck was commissioned to create a full-round view of
Charles I's head to ship to Rome so that Bernini could make a bust
of the king without traveling, he represented the monarch in a neat
architectural combination of views: front, side, and at forty-five
degrees. Why Philippe de Champaigne, given a similarly utilitarian
commission, should have represented Cardinal de Richelieu's bust at
a slight angle, and then in two identical specular profiles, is not
clear. It even seems somewhat wasteful: Richelieu's prominent nose
looks exactly the same when seen from either side.

With the rise of modern science, however, the mensural function of
perspectival images was increasingly challenged by other modes of
projection better suited to technical notation. From the very start,
Alberti had recommended that designers avoid perspective and use instead
a kind of nonforeshortened, scaled drawing-- similar to what today we
would call plans, elevations, and side views in parallel projections.
(11) When drawn in this way, all lines parallel to one another are in
the same scale--a major advantage for technical and construction
drawings. But parallel projections did not exist in the fifteenth
century, and plans, elevations, side views, and sections long remained a
practice without a mathematical theory: The rules of parallel projection
were formalized by French mathematician Gaspard Monge only at the end of
the eighteenth century. Monge's method, known as descriptive
geometry, uses two sets of parallel projections to univocally notate the
position of any point in space onto two planes that, if needed, can be
drawn on the same sheet of paper. (12)

DESCRIPTIVE GEOMETRY is a brilliant mathematical invention, and
when it is put to practical use, its efficacy is formidable. No one
could "store" a full size skyscraper--say, the Seagram
Building--in reality; but many offices could store, in a few drawers,
the batch of project drawings necessary to make it and, if needed, to
remake it. With parallel projections (including axonometric views, which
came a bit later), (13) the art of compressing big 3-D objects onto
small, flat sheets of paper (or parchment or canvas or Mylar) reached
the apex of modern quantitative precision: Parallel projections do not
even try to look like the objects they represent, but aim at recording
and transmitting the measurements, place, and shape of a volume in space
as precisely as possible--and using as little data as possible.
Projections convert volumes (i.e., a number of points equal to infinity
to the power of three) into surfaces (a number of points equal to
infinity to the power of two). In pure data metrics, one could thus say
that the savings are almost infinite--or at least very big. However,
such data cheapness is increasingly unwarranted today: Using digital
technologies, we can already store not only a huge number of planar
drawings but also full 3-D avatars of buildings on a single memory
chip--including all the data we need to simulate that building in
virtual reality, or to build it in full.

Oddly, even this latest technological leap was anticipated by
Alberti himself. In his mid-fifteenth-century treatise De statua (On
Sculpture), Alberti had introduced a revolutionary 3-D design and
fabrication method, entirely based on digital data. Thanks to a peculiar
measuring device of his invention (a wheel with a revolving spoke and
hanging plumb lines), Alberti claimed that free-standing, solid bodies
could be scanned, recorded, transmitted, and replicated, in full or in
part, exclusively using numbers--and without any recourse to images.
(14) This, however, would have required the manual processing of an
extraordinary number of measurements; hence Alberti's precocious
cad/cam technology immediately fell into oblivion. Several equally
unfruitful attempts to develop similar replicating technologies are
recorded at the onset of the mechanical age--including one by Samuel F.
B. Morse, of telegraph fame. Morse, a noted painter, and later a
professor of literature of the arts of design, may have been aware of
Alberti's precedent, but his machine was no more successful than
his Florentine predecessor's. (15)

Both Alberti's and Morse's technologies would have
required a seamless connection between a numerically based scan and a
similarly numerically based fabrication process, which no manual or
mechanical tool can effectively provide. But today's cheap and
increasingly ubiquitous digital 3-D scanners and 3-D printers work
exactly that way.

MANY NOW SEE 3-D printing as a turning point in the history of
technology. At the same time, however, today's rise of 3-D
technologies for scanning and visualization--from the cheap and
versatile Microsoft Kinect, hacked by architectural students around the
world, to more recent developments in stereoscopic virtual reality--is
also likely to change the way we see almost everything, and represent
and know the world around us. Already, 3-D-printed objects seem poised
to take over the role previously played by photographic images in our
daily life. The imagemaking technologies we grew up with would typically
allow us to take a snapshot of any object--say, a cat--and print it out
as a flat, photographic, perspectival picture. But today's
technology allows us to snap a scan of the same cat and print that out
as a sculpture--life-size, if needed. The French company Photomaton
(known for owning and operating thousands of automatic photo booths in
public places), for example, recently launched a 3-D photo booth that
not only produces traditional pictures but also constructs a volumetric
scan of the full figure of the client (a statuette can then be
3-D-printed off-site and shipped to the address provided by the user).
(16) Cheap and affordable technologies such as Autodesk's 123D
Catch and Google's Tango can already turn out 3-D models of large
full-round volumes and of internal spaces, which customers can enrich
with the physical data they need for specific purposes; (17) a voxel
with added information on the properties of the materials that compose
it is often called a maxel, and designers routinely use 3-D models to
render geometry and shapes as well as to simulate all kinds of
performance (structural, thermal, energy, etc.).

Yet whether produced for highly technical tasks or for
general-audience purposes (first and foremost, entertainment and
gaming), 3-D models are in fact very seldom printed. Distributed 3-D
printing may soon upend global manufacturing, but just as most
photographs have long been seen only on electronic displays,
today's 3-D models are best navigated in simulations. The first
consumer light- field camera, the Lytro, was released in 2012; it was
advertised as a camera that allowed customers to refocus and marginally
shift the vantage point of each picture after the snapshot has been
taken. (18) It was not a success, partly due to the limits of
light-field technologies for depth sensing, but the implications of its
new image-making process were vast and momentous: When you take a
picture that way, you do not project in onto a screen (the Albertian
way) once and for all; you create a 3-D model in space that you can
eventually visit at will, looking in different directions and moving
around it (in Albertian terms, rotating the central ray and changing the
point of view). In 2014, ScanLAB Projects (a spinoff of a research group
at the Bartlett School of Architecture in London) (19) recorded an
entire Vivienne Westwood fashion shoot as a volumetric point cloud, and
in the summer of 2016 various sports events were broadcast live in
virtual reality (including some from the Rio Olympics), to be
experienced through head-mounted displays. The degree of immersiveness
supported by these VR technologies is variable--the vantage point of the
end user may be fixed or moving and the angle of rotation of the head
more or less wide; the headsets do not have to be stereoscopic, although
it helps if they are. (20) Alongside virtual reality, a new generation
of head-mounted displays supports augmented-reality and mixed-reality
reenactments. In fact, the ways to exploit and experience a 3-D model,
once it is made, are countless, and planar images still have many
practical advantages over 3-D models: So long as we have eyes to see, we
shall keep using monocular images (better if paired and synced for
stereoscopy) for all kind of reasons and tasks. But the competitive edge
that projected images enjoyed for centuries over 3-D models was due as
much to physical as to data lightness. From Alberti until recently,
projected images were the best way to capture, notate, and replicate all
sorts of 3-D originals, because projections (perspectival or otherwise)
compress a lot of spatial information into small and portable planar
files--most of the time, as small as a piece of paper. That still holds
true, but it matters less and less, because data is now so easy to
gather and so cheap to keep and copy. Soon, we shall use our cell phones
to take 3-D scans, not photographs. (21) And keeping, editing, sending,
navigating, sharing, or even printing a statuary selfie will soon cost
the same as keeping, editing, sending, viewing, sharing, or even
printing a pictorial one.

At the end of the Middle Ages, the conflation of a new technology
for capturing and compressing images and of a new technology for
reproducing them changed the history of the West. Today, the conflation
of new technologies for capturing and reproducing reality directly in
three dimensions, without the mediation of projected images, is likely
to have similarly epochal consequences. In the mid-sixteenth century,
Jacopo da Pontormo, the lunatic Florentine painter, could claim that
while God needed three dimensions to create nature, painters only needed
two to re- create it; which, he concluded, "is truly a miraculous,
divine artifice." (22) Now we need far less of that artifice, as we
can represent and reproduce the world just as it was made-in three
dimensions. Both ekphrasis and projective imagemaking were as much a
virtue as a practical necessity in times of small data. But those times
are over--data is now so cheap and ubiquitous that we no longer need to
skimp on it. Alphabetical notation and projected images are
data-compression technologies that served us well, but which we no
longer need. Three-dimensional models have replaced texts and images as
our tools of choice for the notation, replication, representation, and
quantification of the physical world around us. Once verbal, then
visual, knowledge can now be recorded and transmitted in an entirely new
spatial format.

NOTES

(1.) This essay derives in part from a chapter of Mario
Carpo's forthcoming book, The Second Digital Turn: Design Beyond
Intelligence (Cambridge, MA: MIT Press, 2017), to which the reader is
referred for full citations, translations from the original sources, and
additional biography.

(2.) Pixels are the smallest uniform tiles in a digitized picture;
voxels are the smallest uniform building blocks in a digitized volume.

(3.) The parallel between the discovery of perspective and the
invention of print is as old as Vasari. For some more recent (and
controversial) takes on the matter, see William M. Ivins Jr., Prints and
Visual Communication (Cambridge, MA: Harvard University Press, 1953),
23, 158-80; Friedrich A. Kittler, "Perspective and the Book,"
trans. Sara Ogger, Grey Room, no. 5 (Fall 2001): 38-53. Originally
published in German as "Buch und Perspektive," in Perspektiven
der Buch- und Kommunikationskultur, ed. Joachim Knape and Hermann-Arndt
Riethmuller (Tubingen, Germany: Osiander, 2000), 19-31.

(8.) All the points on the same visual beam, line, or ray (the line
connecting the eye with a point being seen) intersect the picture plane
at the same point, hence they translate into a single point of the
perspectival projection. The mathematical procedure used to extract
actual measurements from a perspectival image, known today as
photogrammetry, has been known since at least the early seventeenth
century: See Filippo Camerota, '"The Eye of the Sun':
Galileo and Pietro Accolti on Orthographic Projections," in
Perspective, Projections & Design: Technologies of Architectural
Representation, ed. Mario Carpo and Frederique Lemerle (London:
Routledge, 2008), 115-25, especially 123.

(9.) By which everyone in the Renaissance meant "perspectival
painting," or the making of perspectival images. That started with
Alberti, who never distinguished between projected images and painting:
His theory of what we now call geometric perspective is set forth in On
Painting, where Alberti in fact never used the term perspective, as if
all images were projections, and all painting perspectival.

(13.) The first rules for drawing what we today call axonometric
views were published by the Cambridge scientist and pedagogist William
Farish in the early 1820s. See Peter Jeffrey Booker, A History of
Engineering Drawing (London: Chatto and Windus, 1963), 114-27.

(14.) Alberti, De statua, in Alberti, On Painting and On Sculpture:
The Latin Texts of "De Pictura" and "De Statua," ed.
and trans. Cecil Grayson (London: Phaidon, 1972), 117-42. De statua was
composed in Latin at some point between 1435 and 1466. Alberti used a
similar number-based technology to scan and copy a map of Rome: See
Mario Carpo and Francesco Furlan, eds., Leon Battista Alberti's
Delineation of the City of Rome (Descriptio vrbis Romee) (Tempe: Arizona
Center for Medieval and Renaissance Texts and Studies, 2007), and Carpo,
The Alphabet and the Algorithm, in particular section 2.2, "Going
Digital," 54-8.

(15.) Samuel F. B. Morse: His Letters and Journals--Edited and
Supplemented by His Son Edward Lind Morse (Boston: Houghton Mifflin,
1914), 1, 245; see also the letter of August 22, 1823, with reference to
his invention of a "machine for sculpture" (247) that would
deliver "perfect copies of any model" (245). See also Morse,
Lectures on the Affinity of Painting with the Other Fine Arts, ed.
Nicolai Cikovsky Jr. (Columbia: University of Missouri Press, 1983), 43,
139.